Geodesic
Bounded oscillation of nonlinear neutral differential equations of arbitrary order
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 1, pp. 185-195 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The paper is concerned with oscillation properties of $n$-th order neutral differential equations of the form \[ [x(t)+cx(\tau (t))]^{(n)}+q(t)f\bigl (x(\sigma (t))\bigr )=0,\quad t\ge t_0>0, \] where $c$ is a real number with $|c|\ne 1$, $q\in C([t_0,\infty ),\mathbb R)$, $f\in C(\mathbb R,\mathbb R)$, $\tau ,\sigma \in C([t_0,\infty ),\mathbb R_+)$ with $\tau (t)$ and $\lim _{t\rightarrow \infty }\tau (t)=\lim _{t\rightarrow \infty }\sigma (t)=\infty $. Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations in which $c$ is a function of $t$ and a certain type of a forcing term is present.
The paper is concerned with oscillation properties of $n$-th order neutral differential equations of the form \[ [x(t)+cx(\tau (t))]^{(n)}+q(t)f\bigl (x(\sigma (t))\bigr )=0,\quad t\ge t_0>0, \] where $c$ is a real number with $|c|\ne 1$, $q\in C([t_0,\infty ),\mathbb R)$, $f\in C(\mathbb R,\mathbb R)$, $\tau ,\sigma \in C([t_0,\infty ),\mathbb R_+)$ with $\tau (t)$ and $\lim _{t\rightarrow \infty }\tau (t)=\lim _{t\rightarrow \infty }\sigma (t)=\infty $. Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations in which $c$ is a function of $t$ and a certain type of a forcing term is present.
Classification : 34K11, 34K40
Keywords: oscillation; positive solutions; neutral equation 
@article{CMJ_2001_51_1_a17,
     author = {Yilmaz, Y. \c{S}ahiner and Zafer, A.},
     title = {Bounded oscillation of nonlinear neutral differential equations of arbitrary order},
     journal = {Czechoslovak Mathematical Journal},
     pages = {185--195},
     year = {2001},
     volume = {51},
     number = {1},
     mrnumber = {1814644},
     zbl = {1079.34540},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_1_a17/}
}
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Yilmaz, Y. Şahiner; Zafer, A. Bounded oscillation of nonlinear neutral differential equations of arbitrary order. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 1, pp. 185-195. http://geodesic.mathdoc.fr/item/CMJ_2001_51_1_a17/